Question

If $P_1$ and $P_2$ be the length of perpendiculars from the origin upon the straight lines $x \sec \theta + y cosec \theta = a$ and $x \cos \theta - y \sin \theta = a \cos 2 \theta$ respectively, then the value of $4P_1{^2} + P_2{^2}$.

• A. $a^2$
• B. $ 2 a^2$
• C. $a^2/ 2$
• D. $3a^2$

Answer 1

Answer: (A)

We have $P_1$ = length of perpendicular from $(0, 0)$ on
$x \sec \theta + y \cos ec \theta = a$
i.e. $P_{1} = \left|\frac{a}{\sqrt{\sec^{2} \theta +\cos ec^{2} \theta}}\right| = \left|a \sin\theta \cos\theta\right| $
$ = \left|\frac{a}{2} \sin 2\theta\right| 2P_{1} = \left|a \sin 2 \theta\right|$
$P_2$= Length of the perpendicular from $(0, 0) $ on
$ x \cos \theta -y \sin\theta =a \cos2\theta $
$ P_{2} = \left|\frac{a \cos 2 \theta}{\sqrt{\cos^{2} \theta+\sin^{2} \theta}}\right| = \left|a \cos2\theta\right| $
Now, $ 4P_{1} ^{2} + P_{2 } ^{2}= a^{2} \sin^{2} 2 \theta + a^{2} \cos^{2} 2\theta = a^{2}. $